Sin(п/4+a)-cos(п/4-a)=0

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вопрос

To solve this trigonometric equation:
sin(π/4 + a) - cos(π/4 - a) = 0

First, we will use the sum and difference identities for sine and cosine:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Applying these identities to the equation, we get:
(sin(π/4)cos(a) + cos(π/4)sin(a)) - (cos(π/4)cos(a) - sin(π/4)sin(a)) = 0

Simplifying further:
(√2/2 cos(a) + √2/2 sin(a)) - (√2/2 cos(a) - √2/2 sin(a)) = 0
(√2/2 cos(a) + √2/2 sin(a) - √2/2 cos(a) + √2/2 sin(a)) = 0
2√2/2 sin(a) = 0
√2 sin(a) = 0
sin(a) = 0

Therefore, the solution to the equation sin(π/4 + a) - cos(π/4 - a) = 0 is a = 0 or a = π.